A Dynamical Color-Neutral Pair Phase and Its Central Character

Conjugate representation data need not lose their entire canonical dynamics when color currents cancel. The degeneracy of a fixed reality relation is repaired by retaining an independent gauge-invariant pair phase. The resulting reduction is explicit, but its electric interpretation depends on the actual central character. The spin lift cannot be replaced silently by an exterior tensor action.

This extends the classical current comparison. It concerns ordinary commuting field amplitudes in the stated normal carrier, not anticommuting variables, an identified lepton, charge quantization or a derived mass law.

The Zero-Moment Constraint Has a Two-Dimensional Quotient

Let $z\in\mathbb C^3$ and $w\in(\mathbb C^3)^*$, with SU3 acting by

$$z\mapsto Uz,\qquad w\mapsto\bar U w.$$

The positive same-sign first-order kinetic form has color moment

$$\mu_A=z^\dagger T_Az-w^\dagger T_A^Tw.$$

Vanishing of every traceless component means

$$zz^\dagger-\bar w w^T=cI_3.$$

The left side has rank at most two, so $c=0$. For a nonzero pair, equality of the two rank-one matrices gives

$$z=r u,\qquad w=r e^{i\phi}\bar u,\qquad \|u\|=1,\quad r>0.$$

SU3 acts transitively on unit $u$, with stabilizer SU2. Dividing by that gauge orbit leaves the radial variable and pair phase. The invariant coordinate is

$$B=w^Tz=R e^{i\phi},\qquad R=r^2.$$

Fixing $\phi$ as part of a reality condition discards its canonical partner. Retaining it gives a nondegenerate reduced first-order one-form:

$$\Theta=\frac i2(z^\dagger dz-dz^\dagger z+w^\dagger dw-dw^\dagger w) =-R\,d\phi, \qquad d\Theta=-dR\wedge d\phi.$$

The two Berry terms involving $u$ cancel, but the dynamical pair phase survives. This is the precise distinction from the previously checked fixed conjugation constraint.

Positive Metric and the Cone Coordinate

A horizontal representative shares the phase equally:

$$z=r e^{i\phi/2}u,\qquad w=r e^{i\phi/2}\bar u.$$

Removing gauge-orbit motion from the positive amplitude metric gives

$$ds^2=2dr^2+\frac{r^2}{2}d\phi^2 =\frac{dR^2+R^2d\phi^2}{2R} =\frac{|dB|^2}{2|B|}.$$

With $s=\sqrt{2R}$, this is $ds^2+(s^2/4)d\phi^2$: a flat cone of angle $\pi$, with its singular tip at the zero field. Locally $B=a^2$ is the complex-linear paired-line coordinate of the earlier current construction; $a$ and $-a$ represent the same nonzero gauge orbit. The quotient geometry is a field-value space, not an extra physical spatial sector.

The Actual Lift Decides the Electric Character

The normal degree-one and degree-two exterior components transform as $W$ and $\Lambda^2W\simeq W^*\otimes D$, where $D=\det W$. In that exterior U3 action their central weights are one and two. The invariant wedge therefore lies in $D$ and has weight three.

The actual normal spinor has the half-determinant shift:

$$\Sigma_6=\Lambda^*W\otimes D^{-1/2},\qquad N_{\rm spin}=N-\frac32.$$

The degree-one and degree-two weights are then $-1/2,+1/2$. Their bilinear lies in $D\otimes D^{-1}=\mathbf1$ and is centrally neutral. The local spin lift, and the global existence or cancellation of its line factor, must be respected.

More generally, component weights $q_z,q_w$ give

$$B\mapsto e^{i(q_z+q_w)\theta}B,\qquad Q_{\rm moment}=\frac{q_z+q_w}{3}R$$

when determinant weight three defines one charge unit. The central motion splits into this quotient phase and a true SU3 gauge motion; the checker verifies that decomposition directly.

Thus a specified determinant character can give a color-neutral charged pair, while the unmodified spin lift gives a neutral pair. The reduction itself does not choose between these characters. It also does not fix the continuous action variable $R$ or quantize an absolute charge.

The Pair Clock Is Not Automatically the Particle Clock

For a common positive linear mode clock,

$$H=\omega(\|z\|^2+\|w\|^2)=2\omega R,$$

the original components vary as $e^{-i\omega t}$ while $B$ varies as $e^{-2i\omega t}$. The canonical equations give the same doubled pair phase frequency. Interpreting the bilinear as the observed particle therefore changes which oscillation supplies its mass-frequency. The maintained particle frequency cannot be transferred unchanged merely by forming this invariant. A physical reduction must identify the observed clock and preserve the declared frequency law.

Full Spatial Currents Across the Occupied Sectors

At depths two through four, color acts on the normal factor and every active Clifford matrix acts on the base factor. The common complex-linear paired branch therefore retains arbitrary base spinors and vanishing color currents, as already proved. The Hodge action constrains these currents only at depth two; active three/four also permit ordinary sourced triplets. This quotient does not preferentially select lepton assignments.

At five/six, the residual SU2 has two fundamental copies distinguished by the first active normal occupation. Write their amplitudes as $z$ and $e^{i\phi}\varepsilon\bar z$, with the SU2 alternating matrix $\varepsilon$. Their density and all four base-direction currents cancel. If they share a normalized base spinor $\sigma$, define its base chirality mean

$$s=\sigma^\dagger\Gamma_4\sigma.$$

The actual additional Clifford matrices give the exact obstruction

$$\sum_Aj_{5A}^2=s^2\|z\|^4, \qquad \sum_Aj_{6A}^2=s^2\|z\|^4$$

for the sixth direction when present. A generic density-zero pair thus fails the full spatial source constraints. A specially prepared $s=0$ spinor can evade this test, but the zero-chirality condition and its evolution need to be preserved by the complete spatial dynamics. This is consistent with the earlier canceled positive modes and their packet limitation; it is not a contradiction of the complete residual singlet branches.

At ten there is no independent normal algebra in the selected ten-parent, so this SU3 quotient is not an additional depth-ten constraint. The same Hodge gate is independently recomputed at every occupied depth: two/five/six/ten inactive, three/four active. No calculations at unoccupied depths are needed.

Reproducibility and Use

The checker and report preserve 453 phase, metric, character and full-current checks across 14 cases, plus 192 exact geometry dependency checks. They retain gauge horizontality, a nonzero symplectic form, the actual spin shift, determinant branches, the bilinear frequency and both residual added-direction current norms.

The useful mathematical construction is a nondegenerate gauge-neutral phase derived from paired amplitudes, with an explicit geometric character and metric. A physical charged-lepton mechanism still requires the common electric character, complete occupied spatial dynamics and the correct particle clock. Those are identifiable conditions to derive rather than a reason to discard the dynamical-phase possibility.